• gruppo fondamentale
• geometria algebrica

If E → X is a finite, connected Galois covering, it may be regarded as a principal bundle with fiber the finite group Aut( E/X ), and we may find the profinite completion of the fundamental group as the limit of Aut( E/X ) along the cofiltered category of such coverings. Grothendieck in [Revetements etales et groupe fondamental, 1971] used this strategy to define the étale fundamental group of a scheme using étale coverings.
Lately, Nori in [The fundamental group scheme, 1982] used the same approach replacing étale coverings with principal bundles: this allowed the fibers of the coverings to have a richer structure of affine group schemes. This change is particularly important in characteristic p, where affine group schemes are in general very different from classical groups.
The interesting part of Nori’s theory is the Tannakian interpretation of the fundamental group scheme. When the base X is complete, Nori proved that the category of finite representations of the fundamental group scheme is tensor-equivalent to the category of vector bundles of finite rank on X with a particular finiteness condition, the essentially finite vector bundles.
Following the work of Vistoli and Borne in [The Nori fundamental gerbe of a fibered category, 2012], we simplify Nori’s proof of the Tannakian duality and extend it to pseudo-proper schemes.